Variational Principles from the Schwarz Inequality

The Schwarz inequality is used to generate variational principles for evaluating perturbation sums. By constructing two suitable kets and applying Schwarz's inequality to them, a bound results which contains an arbitrary (variational) operator. The method is illustrated with reference to a family of frequency-dependent sums for which the corresponding bound is shown to be a generalized form of Hylleraas's variation–perturbation principle. It is also pointed out that the Schwarz inequality provides bounds on simple matrix elements.